On Domination Coloring in Graphs

09/09/2019

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by   Yangyang Zhou, et al.

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A domination coloring of a graph G is a proper vertex coloring of G such that each vertex of G dominates at least one color class, and each color class is dominated by at least one vertex. The minimum number of colors among all domination colorings is called the domination chromatic number, denoted by χ_dd(G). In this paper, we study the complexity of this problem by proving its NP-Completeness for arbitrary graphs, and give general bounds and characterizations on several classes of graphs. We also show the relation between dominator chromatic number χ_d(G), dominated chromatic number χ_dom(G), chromatic number χ(G), and domination number γ(G). We present several results on graphs with χ_dd(G)=χ(G).